Service Restoration of Distribution System Considering Novel Battery Charging and Swapping Station, Repair Crews, and Network Reconfigurations

Xianqiu Zhao; Qingshan Xu; Yongbiao Yang

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Service Restoration of Distribution System Considering Novel Battery Charging and Swapping Station, Repair Crews, and Network Reconfigurations PDF

- ORCID：
Xianqiu Zhao ¹
✉
- ORCID：
Qingshan Xu ^1,2
✉
- ORCID：
Yongbiao Yang ^1,2
✉

1. School of Electrical Engineering, Southeast University, Nanjing210096, China； 2. Nanjing Center for Applied Mathematics, Nanjing210096, China

Updated：2024-12-19

DOI：10.35833/MPCE.2024.000010

OUTLINE

Abstract

With the integration of wind power, photovoltaic power, gas turbine, and energy storage, the novel battery charging and swapping station (NBCSS) possesses significant operational flexibility, which can aid in the service restoration of distribution system (DS) during power outages caused by extreme events. This paper presents an integrated optimization model for DS restoration that considers NBCSS, repair crews, and network reconfigurations simultaneously. The objective of this model is to maximize the restored load while minimizing generation costs. To address the uncertainties associated with renewable energies, a two-stage stochastic optimization framework is employed. Additionally, copula theory is also applied to capture the correlation between the output of adjacent renewable energies. The conditional value-at-risk (CVaR) measure is further incorporated into the objective function to account for risk aversion. Subsequently, the proposed optimization model is transformed into a mixed-integer linear programming (MILP) problem. This transformation allows for tractable solutions using commercial solvers such as Gurobi. Finally, case studies are conducted on the modified IEEE 33-bus and 69-bus DSs. The results illustrate that the proposed method not only restores a greater load but also effectively mitigates uncertainty risks.

Keywords

Novel battery charging and swapping station; distribution system; service restoration; two-stage stochastic optimization; conditional value-at-risk

I. Introduction

IN recent years, extreme events such as hurricanes, storms, floods, and cyber-attacks, have caused severe damages to the power grids, resulting in large-scale blackouts and significant economic losses [

1]-[3]. For example, in 2019, the super typhoon “Lekima” struck the southeast coast of China, leading to power outages for more than 7 million customers and an economic loss of totaling 53 billion yuan [4]. As a bridge to link the transmission network and customers, the distribution system (DS) is a vital public infrastructure that ensures the normal operation of the social economy. According to the statistics, the failures of DS are responsible for 90% of the customer outage minutes [5]. Consequently, the development of an effective and efficient distribution service restoration (DSR) strategy stands as one of the paramount tasks for utilities.

Numerous studies have examined the service restoration of DS. Among them, network reconfigurations are popularly employed. For instance, a two-stage network reconfiguration method is proposed for the self-healing scheme of DS in [

6]. Reference [7] introduces a fully decentralized multi-agent system-based network reconfiguration method for DS, which can lessen the computational burden of service restoration. In [8], an hourly basis network reconfiguration is adopted to enhance the resilience of DS.

Nowadays, the increasing penetration of distributed energy resources (DERs) transforms the operation and control paradigms of DS [

9]. In this context, the active DS possesses greater potential for realizing self-healing capability [10]. Photovoltaic (PV) power, wind power (WP), and energy storage (ES) are all considered for the service restoration of DS after wildfire [11]. To cope with the extended events, the adaptive formation of microgrids is proposed for critical service restoration in [12]. Reference [13] presents a sequential DS service restoration method based on microgrid formation in the case of large-scale power outages. Nevertheless, given the presence of isolated fault regions, complete DS restoration may be impeded until damages are cleared. Hence, some studies have introduced the scheduling of repair crews into the service restoration of DS. Reference [14] proposes a coordinated repair and operation model for the service restoration of DS. In [15], the repair crews, line switch operations, mobile power sources, DERs, and ES are coordinated to maximize the restored load of DS. Reference [16] explores a multi-period coordinated repair scheduling and service restoration for DS, considering soft open points (SOPs). These studies have made substantial contributions to the field of DS restoration, enhancing the feasibility and optimization of the restoration process.

Electric vehicles (EVs) have witnessed widespread adoption across the globe. Battery swapping stations (BSSs), offering the capability to swiftly recharge EVs in mere minutes, have garnered increasing attention. BSS can also play a crucial role in supporting the operation of DS through the flexible process to charge and discharge stocked substantial batteries. Surprisingly, scant research has explored the potential of BSS in service restoration of DS. Reference [

17] presents an optimization strategy to improve the resilience of an island microgrid by incorporating BSS in emergency scenarios. Along with network reconfigurations, the BSS is also considered for the service restoration of DS in the case of extreme weather [18]. Reference [19] also proposes a joint service restoration for the DS, where DERs, BBS, and network reconfigurations are included. Nonetheless, these research works come with certain limitations. First, none of them has considered the scheduling of repair crews, whose importance has been highlighted. Furthermore, these research works do not consider the uncertainties of renewable energies, which can significantly impact the restoration process. Additionally, the integration of WP, PV, gas turbine (GT), and ES enables the upgrade of the BSS into a novel battery charging and swapping station (NBCSS), greatly enhancing operational flexibility [20]. The NBCSS holds the potential to further enhance the reliability of service restoration of DS.

The comparison of this paper with the most relevant research works mentioned before is given in Table I. To address the above gaps, this paper proposes a coordination method that integrates crew dispatch with the service restoration of DS while considering the NBCSS. The major contributions of this paper are summarized as follows.

TABLE I Comparison of This Paper with Most Relevant Research Works

Reference	Network reconfiguration	Repair crew	BSS	Uncertainty
[14]	√	√		√
[15]	√	√		√
[16]	√	√
[17]			√	√
[18]	√		√
[19]	√		√
This paper	√	√	√	√

1) An integrated method coordinating NBCSS, repair crews, and network reconfigurations is proposed for the service restoration of DS. The goal of this model is to maximize the restored load while minimizing generation costs.

2) A stochastic optimization framework is employed to cope with the uncertainties of renewable energies, where the potential correlation between the output of adjacent renewable energies is also described by copula theory. Moreover, the conditional value-at-risk (CVaR) measure is included in the objective function for risk aversion.

The rest of the paper is organized as follows. The framework of DS incorporating NBCSS is shown in Section II. Section III gives the model formulation. The two-stage risk-averse stochastic optimization framework is given in Section IV. Case studies are demonstrated in Section V. Finally, Section VI draws the conclusion.

II. Framework of DS Incorporating NBCSS

The framework of DS incorporating NBCSS is illustrated in Fig. 1. The DS comprises renewable energies (WP and PV), load, and lines, while the NBCSS is composed of WP, PV, GT, ES, and charging and discharging equipment (CDE). Then, the NBCSS is connected to the DS at a certain node.

Fig. 1 Framework of DS incorporating NBCSS.

In normal scenarios, the DS can fully meet all load demands, and NBCSS can satisfy all battery requirements. However, during extreme events, the power supply from the upper grid is interrupted, and some distribution lines are damaged, leading to power outages for some users. In such critical situations, the top priority is the service restoration of DS. Before implementing service restoration, fault positioning and assessment are prerequisites. Since this paper focuses on repair and resource scheduling, the locations and repair time of the damages are known from the prerequisites. Hence, with such information, the repair crews are dispatched to repair the damaged lines and restore the outage load, which leave the depot site to repair the damaged lines and return to the depot site once all of the allocated duties have been completed. Meanwhile, the NBCSS suspends its battery swapping services and is arranged to support the load service restoration. In addition, the network configurations are also employed to coordinate the repair crews to reduce the load outage time.

Thus, in this paper, the repair and resource scheduling for DS is co-optimized to minimize the unserved load plus the generation cost by coordinating the NBCSS, repair crews, and network configurations.

III. Model Formulation

A. Model of NBCSS

The state of charge (SOC) interval technique is used to model the batteries in the CDE, as presented in our prior research work [

20]. The modeling details can be found in the reference provided and will not be elaborated here.

1) CDE

\{\begin{array}{l} N_{1, s, t} = N_{1, s, t - 1} - N_{1, c, t} + N_{2, d, t} \\ N_{2, s, t} = N_{2, s, t - 1} + N_{1, c, t} - N_{2, c, t} - N_{2, d, t} + N_{3, d, t} \\ ⋮ \\ N_{K - 1, s, t} = N_{K - 1, s, t - 1} + N_{K - 2, c, t} - N_{K - 1, c, t} - N_{K - 1, d, t} + N_{K, d, t} \\ N_{K, s, t} = N_{K, s, t - 1} + N_{K - 1, c, t} - N_{K, d, t} \end{array}

(1)

N_{a l l, c, t} = \sum_{i = 1}^{K - 1} N_{i, c, t}

(2)

N_{a l l, d, t} = \sum_{i = 2}^{K} N_{i, d, t}

(3)

\{\begin{array}{l} 0 \leq N_{i, c, t} \leq N_{i, s, t - 1} \\ 0 \leq N_{i + 1, d, t} \leq N_{i + 1, s, t - 1} \end{array} i \in 1,2, \dots, K - 1

(4)

\{\begin{array}{l} 0 \leq N_{a l l, c, t} \leq N_{m a x} σ_{c, t} \\ \begin{array}{l} 0 \leq N_{a l l, d, t} \leq N_{m a x} σ_{d, t} \\ σ_{c, t} + σ_{d, t} \leq 1 \end{array} \end{array}

(5)

\{\begin{array}{l} 0 \leq β_{i, c, t} - (N_{a l l, c, t} - \sum_{k = 1}^{i} N_{K - k, s, t}) \\ β_{i, c, t} - (N_{a l l, c, t} - \sum_{k = 1}^{i} N_{K - k, s, t}) \leq M_{b i g} (1 - δ_{i, c, t}) \\ 0 \leq β_{i, c, t} \leq M_{b i g} δ_{i, c, t} \\ N_{K - i, s, t - 1} - M_{b i g} (1 - δ_{i, c, t}) \leq N_{K - i, c, t} \\ N_{K - i - 1, c, t} \leq M_{b i g} δ_{i, c, t} \\ δ_{i + 1, c, t} \leq δ_{i, c, t} \end{array} i \in 1,2, \dots, K - 2

(6)

\{\begin{array}{l} 0 \leq α_{i, c, t} - (N_{a l l, d, t} - \sum_{k = 1}^{i} N_{k + 1, s, t}) \\ α_{i, c, t} - (N_{a l l, d, t} - \sum_{k = 1}^{i} N_{k + 1, s, t}) \leq M_{b i g} (1 - θ_{i, c, t}) \\ 0 \leq α_{i, c, t} \leq M_{b i g} θ_{i, c, t} \\ N_{i + 1, s, t - 1} - M_{b i g} (1 - θ_{i, c, t}) \leq N_{i + 1, d, t} \\ N_{i + 2, d, t} \leq M_{b i g} θ_{i, c, t} \\ θ_{i + 1, c, t} \leq θ_{i, c, t} \end{array} i \in 1,2, \dots, K - 2

(7)

\{\begin{matrix} P_{a l l, c, t} = N_{a l l, c, t} P_{f i x, c} \\ P_{a l l, d, t} = N_{a l l, d, t} P_{f i x, d} \end{matrix}

(8)

Equation (1) gives the quantitative relationship of batteries for different SOC intervals. The limitations on the overall amounts of charging and discharging batteries are given in (2) and (3). The restrictions on the amount of charging and discharging batteries, which are determined by the SOC intervals, are presented in (4). Equation (5) gives mutually exclusive limitations for charging and discharging batteries. Equations (6) and (7) demonstrate the charging and discharging priorities determined by the big-M method. Equation (8) states the total power of charging and discharging batteries.

2) GT

x_{g, t} - x_{g, t - 1} = u_{g, t} - v_{g, t}

(9)

u_{g, t} + v_{g, t} \leq 1

(10)

u_{g, t} \leq x_{g, τ} \forall t \leq τ \leq t + t_{g}^{U} - 1

(11)

v_{g, t} \leq 1 - x_{g, τ} \forall t \leq τ \leq t + t_{g}^{D} - 1

(12)

P_{g, m i n} x_{g, t} \leq P_{g, t} \leq P_{g, m a x} x_{g, t}

(13)

- S_{g, m a x} x_{g, t} \leq Q_{g, t} \leq S_{g, m a x} x_{g, t}

(14)

P_{g, t} - P_{g, t - 1} \leq R_{U, g} x_{g, t - 1} Δ t + S_{U, g} u_{g, t} Δ t

(15)

P_{g, t - 1} - P_{g, t} \leq R_{D, g} x_{g, t} Δ t + S_{D, g} v_{g, t} Δ t

(16)

P_{g, t}^{2} + Q_{g, t}^{2} \leq S_{g, m a x}^{2}

(17)

Equations (9) and (10) express the start-up and shut-down relationships. Equations (11) and (12) present the minimum up-time and down-time restrictions. The active and reactive power constraints are ensured in (13) and (14), while the ramp-up and ramp-down restrictions are given in (15) and (16). Equation (17) shows that the active power and reactive power are restricted by the maximum capacity of apparent power.

3) ES

E_{e, t} = \{\begin{array}{l} E_{e, 0} & t = 0 \\ E_{e, t - 1} + (η_{c h, e} P_{c h, t} - \frac{P_{d i s, t}}{η_{d i s, e}}) Δ t & t \geq 1 \end{array}

(18)

λ_{e, m i n} E_{e, r a t e} \leq E_{e, t} \leq λ_{e, m a x} E_{e, r a t e}

(19)

E_{e, T} = E_{e, 0}

(20)

0 \leq P_{c h, t} \leq P_{c h, m a x} σ_{c h, t}

(21)

0 \leq P_{d i s, t} \leq P_{d i s, m a x} σ_{d i s, t}

(22)

σ_{c h, t} + σ_{d i s, t} \leq 1

(23)

Equations (18)-(20) illustrate the energy constraints of ES, while (21)-(23) specify the constraints of charging and discharging power of ES.

4) Restored Load

0 \leq P_{t}^{N, L} \leq P_{t}^{N, L, m a x}

(24)

0 \leq Q_{t}^{N, L} \leq Q_{t}^{N, m a x}

(25)

Q_{t}^{N, L, m a x} P_{t}^{N, L} = Q_{t}^{N, L} P_{t}^{N, L, m a x}

(26)

Equations (24) and (25) give the limits of restored load. Equation (26) demonstrates a fixed ratio between restored active and reactive power.

5) Renewable Energies

1) WP

0 \leq P_{t}^{N, W P} \leq P_{t, R E}^{N, W P}

(27)

(P_{t}^{N, W P})^{2} + (Q_{t}^{N, W P})^{2} \leq (S_{m a x}^{N, W P})^{2}

(28)

2) PV

0 \leq P_{t}^{N, P V} \leq P_{t, R E}^{N, P V}

(29)

(P_{t}^{N, P V})^{2} + (Q_{t}^{N, P V})^{2} \leq (S_{m a x}^{N, P V})^{2}

(30)

Equations (27) and (29) show that the active power output of renewable energies is limited by the maximum available value. Equations (28) and (30) indicate that the active power and reactive power are restricted by the maximum capacity of apparent power.

6) Exchanging Power

P_{j, t}^{N, e x} = P_{g, t} + P_{d i s, t} - P_{c h, t} + P_{t}^{N, W P} + P_{t}^{N, P V} - P_{t}^{N, L} + P_{a l l, d, t} - P_{a l l, c, t}

(31)

Q_{j, t}^{N, e x} = Q_{g, t} + Q_{t}^{N, W P} + Q_{t}^{N, P V} - Q_{t}^{N, L}

(32)

- P_{j, t}^{N, e x, m a x} \leq P_{j, t}^{N, e x} \leq P_{j, t}^{N, e x, m a x}

(33)

- Q_{j, t}^{N, e x, m a x} \leq Q_{j, t}^{N, e x} \leq Q_{j, t}^{N, e x, m a x}

(34)

Equations (31) and (32) give the active and reactive power of NBCSS exchanged with DS. Equations (33) and (34) give the maximum limits of exchanged active and reactive power.

B. Scheduling of Repair Crews

The optimal scheduling of repair crews can be modeled by using the graph theory [

21]. Its goal is to seek the best route for each repair crew to reach the damaged components. Assume there are N damaged lines in an extreme event and define the node set of damaged lines

Φ_{F}

{1,2, \dots, N}

with the depot node as {0}. Then, the route table, which is

N + 1

dimension square matrix composed of binary entries, is employed to properly portray the crew routing alternative. Each repair crew leaves from depot node 0 to repair damaged lines and ultimately returns to the depot node 0. Figure 2 illustrates the crew route decision and its associated route table value. The constraints related to the crew route decision are presented as below.

Fig. 2 Illustration of crew route decision and its associated route table value.

\sum_{i \in Φ_{F}} x_{0, i, c} = 1 \forall c

(35)

\sum_{i \in Φ_{F}} x_{i, 0, c} = 1 \forall c

(36)

\sum_{j \in Φ_{F}} x_{j, i, c} - \sum_{k \in Φ_{F}} x_{i, k, c} = 0 \forall c, \forall i

(37)

R_{i, c} = \sum_{j \in Φ_{F}} x_{j, i, c} \forall c, \forall i

(38)

\sum_{c} R_{i, c} = 1 \forall i

(39)

T_{0, c} = t_{0} \forall c

(40)

T_{i, c} - T_{0, c} \leq t_{0, i, c}^{t r i p} + M_{b i g} (1 - x_{j, i, c}) \forall c

(41)

t_{0, i, c}^{t r i p} - M_{b i g} (1 - x_{0, i, c}) \leq T_{i, c} - T_{0, c} \forall c

(42)

T_{i, c} - T_{j, c} \leq t_{j, i, c}^{t r i p} + t_{j, c}^{r e p} + M_{b i g} (1 - x_{j, i, c}) \forall c, \forall i, j \in Φ_{F}

(43)

t_{j, i, c}^{t r i p} + t_{j, c}^{r e p} - M_{b i g} (1 - x_{j, i, c}) \leq T_{i, c} - T_{j, c} \forall c, \forall i, j \in Φ_{F}

(44)

0 \leq T_{i, c} \leq M_{b i g} R_{i, c} \forall i \in Φ_{F}

(45)

\sum_{t} f_{i, t} = 1 \forall i \in Φ_{F}

(46)

\sum_{t} t f_{i, t} \leq \sum_{c} (T_{i, c} + t_{j, c}^{r e p} R_{i, c}) + 1 - ε \forall i \in Φ_{F}

(47)

\sum_{c} (T_{i, c} + t_{j, c}^{r e p} R_{i, c}) \leq \sum_{t} t f_{i, t} \forall i \in Φ_{F}

(48)

y_{i, t} \leq \sum_{τ = 1}^{t - 1} f_{i, τ} \forall i \in Φ_{F}

(49)

Equations (35) and (36) enforce that each crew starts and ends its routes at depot node 0. Equation (37) implies that each crew repairs the damaged lines sequentially. Equations (38) and (39) assume that each damaged line is repaired by only one crew and only once. Equation (40) gives the time when each repair crew leaves the depot node. Equations (41) and (42) present the relationship between the arrival time at damaged line i and the departure time from the depot node. Equations (43) and (44) also imply the relationship of arrival time between damaged lines i and j. Equation (45) limits the value of the arrival time of each crew. Equations (46)-(48) calculate the repair completion time of damaged lines i. Equation (49) gives the relationship between the repair state and the available state of damaged line i. It means that the damaged lines are available right away once they are fixed.

C. Fault Isolation and Network Reconfigurations

When a line is damaged, fault isolation is required to ensure the normal operation of the remaining parts [

22]. Moreover, network reconfigurations are also used to improve the operational flexibility of DS. The relevant constraints are given as:

ε_{i j, t} = 1 \forall (i, j) \notin {Ω_{D} ⋃ Ω_{S}}

(50)

ε_{i j, t} \leq y_{k (i j), t} \forall i, \forall j

(51)

ε_{h s, t} \leq y_{k (i j), t} \forall i, \forall j, \forall (h, s) \in W_{i j}^{U} ⋃ W_{i j}^{D}

(52)

\sum_{(i, j) \in Β_{l}} ε_{i j, t} \leq Β_{l} - 1 \forall l \in O_{l}

(53)

Equation (50) defines the default status of the nondamaged and nonswitchable lines. Equation (51) shows that the switch state of the damaged lines is restricted by the corresponding available state. Equation (52) achieves fault isolation by restricting the switch state of the upstream and downstream switches. Equation (53) is the radiality constraint, which ensures that each potential loop in the DS has at least one line open.

D. Model of DS

The linearized DistFlow branch model is used to model the DS [

23].

P_{j, t}^{N, e x} + P_{j, t}^{D, W P} + P_{j, t}^{D, P V} + P_{i j, t}^{D} = \sum_{k \in S_{j}^{B, d}} P_{j k, t}^{D} + P_{j, t}^{D, L}

(54)

Q_{j, t}^{N, e x} + Q_{j, t}^{D, W P} + Q_{j, t}^{D, P V} + Q_{i j, t}^{D} = \sum_{k \in S_{j}^{B, d}} Q_{j k, t}^{D} + Q_{j, t}^{D, L}

(55)

V_{i, t}^{} - V_{j, t}^{} \leq (P_{i j, t}^{D} r_{i j}^{D} + Q_{i j, t}^{D} x_{i j}^{D}) / V_{0}^{} + M_{b i g} (1 - ε_{i j, t})

(56)

(P_{i j, t}^{D} r_{i j}^{D} + Q_{i j, t}^{D} x_{i j}^{D}) / V_{0}^{} - M_{b i g} (1 - ε_{i j, t}) \leq V_{i, t}^{} - V_{j, t}^{}

(57)

V_{i, m i n}^{} \leq V_{i, t}^{} \leq V_{i, m a x}^{}

(58)

- P_{i j, t}^{D, m a x} ε_{i j, t} \leq P_{i j, t}^{D} \leq P_{i j, t}^{D, m a x} ε_{i j, t}

(59)

- Q_{i j, t}^{D, m a x} ε_{i j, t} \leq Q_{i j, t}^{D} \leq Q_{i j, t}^{D, m a x} ε_{i j, t}

(60)

0 \leq P_{j, t}^{D, L} \leq P_{j, t}^{D, L, m a x}

(61)

0 \leq Q_{j, t}^{D, L} \leq Q_{j, t}^{D, L, m a x}

(62)

Q_{j, t}^{D, L, m a x} P_{j, t}^{D, L} = P_{j, t}^{D, L, m a x} Q_{j, t}^{D, L}

(63)

0 \leq P_{i, t}^{D, W P} \leq P_{i, t, R E}^{D, W P}

(64)

(P_{i, t}^{D, W P})^{2} + (Q_{i, t}^{D, W P})^{2} \leq (S_{i, m a x}^{D, W P})^{2}

(65)

0 \leq P_{i, t}^{D, P V} \leq P_{i, t, R E}^{D, P V}

(66)

(P_{i, t}^{D, P V})^{2} + (Q_{i, t}^{D, P V})^{2} \leq (S_{i, m a x}^{D, P V})^{2}

(67)

Equations (54) and (55) represent the nodal balance constraints of active and reactive power. Equations (56) and (57) show the relationship between the magnitude of nodal voltage and the flow of branch power. Equation (58) sets the limits on nodal voltage magnitudes. Equations (59) and (60) show that branch power flows are restricted by their capacities and switch states. Equations (61) and (62) give the limits of restored power at node j. Equation (63) demonstrates a fixed ratio between restored active and reactive power at bus j. Similarly, (64)-(67) give the constraints of the active and reactive power of renewable energies in the DS.

E. Objective Function

The objective function is expressed by (68)-(70). We define that f₁ is the total priority-weighted energy not served; f₂ is the start-up, shut-down, and generation costs of GT; $ω_{j}$ and $ω_{N}$ are the load weights; and $λ_{1}$ and $λ_{2}$ are the weighting factors used to link the two sub-objective functions. In order not to encumber the load restoration, $λ_{1}$ is set to be bigger than $λ_{2}$ .

f = m i n {λ_{1} f_{1} + λ_{2} f_{2}}

(68)

f_{1} = \sum_{t} [\sum_{i} ω_{j} (P_{j, t}^{D, L, m a x} - P_{j, t}^{D, L}) + ω_{N} (P_{t}^{N, m a x} - P_{t}^{N, L})]

(69)

f_{2} = \sum_{t} (C^{S U} u_{g, t} + C^{S D} v_{g, t} + C_{g} P_{g, t})

(70)

IV. Two-stage Risk-averse Stochastic Optimization Framework

A. Copula-based Correlation Description

The copula function is utilized to characterize the correlation between the output of adjacent renewable energies (e.g.,WP and PV). It can effectively capture the relationship between the marginal distribution and joint distribution of random variables.

As stated by Sklar’s theorem [

24], any joint cumulative distribution function

F (X_{1}, X_{2}, \dots, X_{M})

can be expressed as marginal distribution functions

{F_{1} (X_{1}), F_{2} (X_{2}), \dots, F_{M} (X_{M})}

and a copula function

C (\cdot)

, as shown in (71).

F (X_{1}, X_{2}, \dots, X_{M}) = C (F_{1} (X_{1}), F_{2} (X_{2}), \dots, F_{M} (X_{M}))

(71)

Differentiating (71), the joint probability distribution function $f (X_{1}, X_{2}, \dots, X_{M})$ can be obtained as:

f (X_{1}, X_{2}, \dots, X_{M}) = c (F_{1} (X_{1}), F_{2} (X_{2}), \dots, F_{M} (X_{M})) \prod_{i = 1}^{M} f_{i} (X_{i})

(72)

where $f_{i} (X_{i})$ is the probability density function of function $F_{i} (X_{i})$ ; and $c (F_{1} (X_{1}), F_{2} (X_{2}), \dots, F_{M} (X_{M}))$ is the copula density function as:

c (F_{1} (X_{1}), F_{2} (X_{2}), \dots, F_{M} (X_{M})) = \frac{\partial C (F_{1} (X_{1}), F_{2} (X_{2}), \dots, F_{M} (X_{M}))}{\partial (F_{1} (X_{1})) \partial (F_{2} (X_{2})) \dots \partial (F_{M} (X_{M}))}

(73)

Various copula functions such as Gaussian Copula, Gumbel Copula, and Frank Copula have been constructed to reveal the dependence of uncertain variables. In this paper, the multivariate Gaussian copula function is utilized to explore the correlation, as expressed in (74).

C_{G a u} = Φ_{ρ} (Φ^{- 1} (F_{1} (X_{1})), Φ^{- 1} (F_{2} (X_{2})), \dots, Φ^{- 1} (F_{M} (X_{M})))

(74)

With the multivariate Gaussian copula function, we can generate the corresponding scenarios where the correlation is considered.

B. Two-stage Stochastic Optimization

In this subsection, we adopt the two-stage stochastic optimization method to cope with the uncertainties of renewable energies. First, according to the probability distribution and correlation of prediction error, the copula-based method mentioned above is used to generate a large number of random scenarios. Then, the scenario reduction method such as K-medoids clustering is applied to select representative scenarios for lessening the calculative burden [

25]. For

N_{s}

representative scenarios, the corresponding objective function is presented below with

φ_{s}

as the probability of the

s^{t h}

scenario.

m i n \sum_{s} φ_{s} (λ_{1} f_{1, s} + λ_{2} f_{2, s})

(75)

f_{1, s} = \sum_{t} [\sum_{i} ω_{j} (P_{j, t}^{D, L, m a x} - P_{j, t, s}^{D, L}) + ω_{N} (P_{t}^{N, m a x} - P_{t, s}^{N, L})]

(76)

f_{2, s} = \sum_{t} (C^{S U} u_{g, t} + C^{S D} v_{g, t} + C_{g} P_{g, t, s})

(77)

Similarly, the corresponding constraints need to be updated. Considering the content redundancy, they are omitted.

Note that x is the first-stage variable; $u_{s}$ is the value of scenario s; $y_{s}$ is the second-stage variables in scenario s; and A, B, C, D, b, c, d and eare the corresponding matrixes and vectors. The mode is recasted as the compact matrix form as below. The definitions of all variables can be found in [

20].

\underset{x}{m i n} \{c^{T} x + \sum_{s \in N_{s}} φ_{s} d^{T} y_{s}\}

(78)

s.t.

A x \leq b

(79)

B x + C y_{s} + D u_{s} \leq e \forall s

(80)

\begin{array}{l} x = [N_{n e w, t}, N_{i, s, t}, N_{a l l, c, t}, P_{a l l, c, t}, δ_{i, c, t}, β_{i, c, t}, N_{a l l, d, t}, P_{a l l, d, t}, σ_{c, t}, σ_{d, t}, \\ α_{i, c, t}, θ_{i, c, t}, u_{g, t}, v_{g, t}, x_{g, t}, x_{i, j, c}, R_{i, c}, T_{i, c}, f_{i, t}, y_{i, t}, ε_{i j, t}] \end{array}

(81)

u_{s} = [P_{i, t, s, R E}^{D, W P}, P_{i, t, s, R E}^{D, P V}, P_{t, s, R E}^{N, W P}, P_{t, s, R E}^{N, P V}]

(82)

\begin{array}{l} y_{s} = {P_{g, t, s}, P_{t, s}^{N, W P}, Q_{t, s}^{N, W P}, P_{t, s}^{N, P V}, Q_{t, s}^{N, P V}, P_{t, s}^{N, L}, Q_{t, s}^{N, L}, E_{e, t, s}, \\ P_{c h, t, s}, P_{d i s, t, s}, σ_{c h, t, s}, σ_{d i s, t, s}, P_{j, t, s}^{N B C S S}, Q_{j, t, s}^{N B C S S}, P_{i j, t, s}^{D}, Q_{i j, t, s}^{D}, \\ P_{j, t, s}^{D, W P}, Q_{j, t, s}^{D, W P}, P_{j, t, s}^{D, P V}, Q_{j, t, s}^{D, P V}, P_{j, t, s}^{D, L}, Q_{j, t, s}^{D, L}, V_{i, t, s}^{}} \end{array}

(83)

C. Risk Management

The optimal solution x^* is derived based on the optimization problem (78)-(83). Its goal is to minimize the expectation of all representative scenarios, presenting a risk-neutral style. This means the results may be optimistic and lead to excessive losses in some unfavorable scenarios. For some operators, they are risk averse and do not want to bear too much loss. Therefore, to manage the risk brought by uncertainties, the CVaR index [

26] is the utilized in this paper, as expressed by (84). Note that

ζ

is value-at-risk (VaR);

α

is the confidential level;

f_{s}

is the loss function of scenario s; and

{[\cdot]}^{+} = m a x {0, \cdot}

C V a R = \underset{ζ, f_{s}}{m i n} \{ζ + \frac{1}{1 - α} \sum_{s \in N_{s}} φ_{s} [f_{s} {- ζ]}^{+}\}

(84)

With CVaR considered, the objective function is modified as follows. Note that $ρ_{s}$ is an auxiliary variable; and $κ$ is the weighting factor of CVaR.

\underset{x}{m i n} \{c^{T} x + \sum_{s \in N_{s}} φ_{s} d^{T} y_{s} + κ (ζ + \frac{1}{1 - α} \sum_{s \in N_{s}} φ_{s} ρ_{s})\}

(85)

ρ_{s} \geq d^{T} y_{s} - ζ

(86)

ρ_{s} \geq 0

(87)

Two linear inequalities are introduced to replace the nonlinear term $[f_{s} {- ζ]}^{+}$ .

D. Linear Approximation

To improve the solving efficiency, the linear approximation method is adopted to replace the quadratic constraints of (17), (28), (30), (65), and (67), which is prevalent in engineering applications [

27].

\{\begin{array}{l} P_{g, t} + Q_{g, t} \leq \sqrt[]{2} S_{g, m a x} \\ P_{g, t} - Q_{g, t} \leq \sqrt[]{2} S_{g, m a x} \end{array}

(88)

\{\begin{array}{l} P_{t}^{N, W P} + Q_{t}^{N, W P} \leq \sqrt[]{2} S_{m a x}^{N, W P} \\ P_{t}^{N, W P} + Q_{t}^{N, W P} \leq \sqrt[]{2} S_{m a x}^{N, W P} \\ - S_{m a x}^{N, W P} \leq Q_{t}^{N, W P} \leq S_{m a x}^{N, W P} \end{array}

(89)

\{\begin{array}{l} P_{t}^{N, P V} + Q_{t}^{N, P V} \leq \sqrt[]{2} S_{m a x}^{N, P V} \\ P_{t}^{N, P V} - Q_{t}^{N, P V} \leq \sqrt[]{2} S_{m a x}^{N, P V} \\ - S_{m a x}^{N, P V} \leq Q_{t}^{N, P V} \leq S_{m a x}^{N, P V} \end{array}

(90)

\{\begin{array}{l} P_{i, t}^{D, W P} + Q_{i, t}^{D, W P} \leq \sqrt[]{2} S_{i, m a x}^{D, W P} \\ P_{i, t}^{D, W P} - Q_{i, t}^{D, W P} \leq \sqrt[]{2} S_{i, m a x}^{D, W P} \\ - S_{i, m a x}^{D, W P} \leq Q_{i, t}^{D, W P} \leq S_{i, m a x}^{D, W P} \end{array}

(91)

\{\begin{array}{l} P_{i, t}^{D, P V} + Q_{i, t}^{D, P V} \leq \sqrt[]{2} S_{i, m a x}^{D, P V} \\ P_{i, t}^{D, P V} - Q_{i, t}^{D, P V} \leq \sqrt[]{2} S_{i, m a x}^{D, P V} \\ - S_{i, m a x}^{D, P V} \leq Q_{i, t}^{D, P V} \leq S_{i, m a x}^{D, P V} \end{array}

(92)

In summary, the original problem is transformed into a mixed-integer linear programming, which can be tractably solved by off-the-shelf solvers such as Gurobi.

V. Case Studies

In this section, case studies are employed to test the effectiveness and performance of the proposed method. The optimization problem is solved using the GUROBI solver utilizing MATLAB and YALMIP.

A. System Parameters

The modified IEEE 33-bus DS [

28] is used to verify the proposed method. The topology structure of modified IEEE 33-bus DS is shown in Fig. 3. The test system includes three remote-controlled switches on lines 12-22, 18-33, and 25-29. The NBCSS connects to the test system via bus 10. Additionally, two PVs are installed on buses 15 and 28, and a WP is installed on bus 5. T is set to be 24 hours with

Δ t

equaling 1 hour. During an extreme event, six lines (L1-L6) are damaged, and the DS is disconnected from the upper grid, as shown in Fig. 3. There is only one depot site in the DS. Then, two repair crews are arranged to repair the damaged lines. The repair time of the damaged lines is listed in Table II. The travel time between damaged lines ranges from 0 to 1 hour. The main system parameters are shown in Table III, where

S O C_{0}

S O C_{7}

are defined by [20]. Critical load in DS is located at buses 4, 10, 13, 21, 24 and 27. They should be prioritized for power supply. Hence, the weight of the critical load is set to be 3, and the weight of the normal load is set to be 1. Figure 4 gives the day-ahead predicted power of WP, PV, and load in DS and NBCSS. Figure 5 depicts the initial number of batteries categorized by different SOC intervals in the NCBSS. To account for the uncertainties of renewable energies, the standard deviation of WP and PV is set to be 15% and 20% of their mean value, respectively. Additionally, both WPs and PVs have their correlation coefficients set at 0.8. Then, 1000 scenarios are first generated and reduced to 10 representative ones using the K-medoids clustering method. In the objective function, weighting factors

λ_{1}

and

λ_{2}

are set to be 1 and 0.8, respectively. With risk aversion considered,

κ

and

α

are set to be 1 and 0.9, respectively.

Fig. 3 Topology structure of modified IEEE 33-bus DS.

TABLE II Repair Time of Damaged Lines

Crew	Repair time (hour)
Crew	L1	L2	L3	L4	L5	L6
Crew 1	3	4	3	4	3	3
Crew 2	3	3	4	3	4	3

TABLE III System Parameters

Parameter	Value	Parameter	Value
$P_{g, m i n}$	500 kW	$C^{S D}$	¥70
$P_{g, m a x}$	2000 kW	$C_{g}$	0.9 ¥/kWh
$t_{g}^{U}$	4 hour	$V_{i, m i n}^{D}$	0.93
$t_{g}^{D}$	4 hour	$V_{i, m a x}^{D}$	1.07
$R_{U, g}$	1000 kW	$V_{0}^{D}$	1
$R_{D, g}$	1000 kW	K	7
$S_{U, g}$	1000 kW	N_all	700
$S_{D, g}$	1000 kW	N_max	300
$η_{c h, e}$	0.93	P_fix,c	5 kW
$η_{d i s, e}$	0.92	P_fix,d	4.5 kW
$P_{c h, m a x}$	1000 kW	[ $S O C_{0}$ , $S O C_{1}$ ]	[0.2, 0.3]
$P_{d i s, m a x}$	1000 kW	[ $S O C_{1}$ , $S O C_{2}$ ]	[0.3, 0.4]
$E_{e, 0}$	1500 kWh	[ $S O C_{2}$ , $S O C_{3}$ ]	[0.4, 0.5]
$E_{e, r a t e}$	3000 kWh	[ $S O C_{3}$ , $S O C_{4}$ ]	[0.5, 0.6]
$λ_{e, m i n}$	0.2	[ $S O C_{4}$ , $S O C_{5}$ ]	[0.6, 0.7]
$λ_{e, m a x}$	0.9	[ $S O C_{5}$ , $S O C_{6}$ ]	[0.7, 0.8]
$C^{S U}$	¥60	[ $S O C_{6}$ , $S O C_{7}$ ]	[0.8, 0.9]

Fig. 4 Day-ahead predicted power of WP, PV and load in DS and NBCSS. (a) DS. (b) NBCSS.

Fig. 5 Initial number of batteries with different SOC intervals.

B. Optimization Results

The routes of repair crews are shown in Fig. 6.

Fig. 6 Routes of repair crews.

Crew 1 is first dispatched from the depot site to the damaged line L3, and then L5 and L6. Crew 2 is first dispatched from the depot site to damaged line L1 and then L2 and L4. Since the critical load at buses 21 and 24 is affected by damaged lines L1 and L3, the crews repair them preferentially. The repair completion and available time of damaged lines are also given in Table IV. Corresponding to the repair routes, the repair completion time of damaged lines L1 and L3 is the shortest (both are 3.5 hours). The two lines can be available at hour 5. The repair completion time of damaged lines L2 and L5 is 6.8 hours and 7.1 hours, respectively. They can be available at hour 8 and hour 9, respectively. The damaged lines L4 and L6 are repaired at hour 10 and hour 10.6, respectively. Then, all damaged lines can be repaired at hour 12.

TABLE IV Repair Completion and Available Time of Damaged Lines

Damaged line	Repair completion time (hour)	Available time (hour)
L1	3.5	5
L2	6.8	8
L3	3.5	5
L4	10.0	11
L5	7.1	9
L6	10.6	12

The status of remote-controlled switches are presented in Table V. From hour 1 to hour 4, the switch 12-22 is closed to supply power for the load at bus 22. From hour 5 to hour 24, all switches are closed to reconfigure the system. Additionally, isolated microgrids can be formed to restore the load. For example, Fig. 7 gives the topology of microgrid formulation at hour 1. After faults occur, two microgrids are formed with the help of remote-controlled switches. Microgrid 1 is responsible for supplying power to the load at buses 2, 3, 4, 5, and 19. Microgrid 2 is responsible for supplying power to the load at buses 10, 11, 12, 13, 14, 15, and 22.

TABLE V Statuses of Remote-controlled Switches

Time	Status of remote-controlled switch
Time	12-22	18-33	25-29
Hour 1 to hour 4	1	0	0
Hour 5 to hour 24	1	1	1

Fig. 7 Topology of microgrid formulation.

The operation status of GT in NBCSS is shown in Fig. 8. During the periods of hour 1 to hour 15, the GT is turned off. Then, the GT is turned on to supply the insufficient power at hour 16 to hour 24. The power of CDE in NBCSS is also presented in Fig. 9. It is primarily charged during the periods of hour 6 to hour 13 and discharged during the periods of hour 1 to hour 5 and hour 16 to hour 24. Therefore, the mismatch between the load and renewable energies can be reduced.

Fig. 8 Operation status of GT in NBCSS.

Fig. 9 Power of CDE in NBCSS.

Moreover, Fig. 10 also presents the optimal results in the second stage for one of the possible scenarios. Corresponding to the operation status, GT generates electricity during the periods of hour 16 to hour 24. Besides, the NBCSS provides power for DS during the periods of hour 1 to hour 5 and hour 16 to hour 24. By contrast, the surplus power generated by renewable energies is absorbed by the NBCSS during the periods of hour 6 to hour 15. The NBCSS helps the DS to restore more load and reduce the curtailment of renewable energies.

Fig. 10 Optimal results in the second stage.

C. Impact of Network Reconfigurations, NBCSS, and BSS

The impact of network reconfigurations, NBCSS, and BSS on service restoration is analyzed in the next four cases. Case 1 is the proposed method. Case 2 does not consider the network reconfiguration, while case 3 does not consider the NBCSS. Case 4 only considers the BSS. The total restored load of DS is shown in Table VI. It can be observed that the total restored load of the proposed method is the biggest. Meanwhile, the total restored load of case 3 is much less than that of case 2. It demonstrates that the NBCSS has a greater impact on service restoration than network reconfiguration. Compared with case 3, the total restored load of case 4 has been greatly increased with the help of BSS. However, its total restored load is not as high as case 1. It proves that the operational flexibility of NBCSS is greater than that of BSS. Besides, the restored load at different time is also given in Fig. 11. Before all the damaged lines are repaired, some load can be restored with the assistance of network reconfiguration. Therefore, the restored load of case 1 is more than that of case 2 during hour 1 to hour 12. After all the damaged lines are repaired, case 1 and case 2 have the same restored load, while case 4 has a lower restored load. Moreover, without the support of NBCSS or BSS, the restored load of case 3 is particularly less than that of case 1, case 2, and case 4 during the periods of hour 15 to hour 24. This underscores the effective service restoration achieved through the coordination of network reconfiguration and NBCSS.

TABLE VI Total Restored Load over All Time Periods

Case	Total restored load (kWh)
Case 1	85059
Case 2	83482
Case 3	69929
Case 4	82298

Fig. 11 Restored load at different time.

D. Comparison with Deterministic Optimization

Out-of-sample analyses are used to compare the performance of the proposed method and deterministic optimization. The deterministic optimization, as detailed in section III, completely ignores the prediction error. The day-ahead optimization results are first obtained based on the corresponding methods. Following this, the entire optimization problem is solved using 1000 new out-of-sample scenarios based on the day-ahead results. The computational performance for the two methods is summarized in Table VII. In terms of both the average and worst-case values of the total restored load, the proposed method consistently outperforms deterministic optimization. This outcome emphasizes the necessity of fully accounting for uncertainties related to renewable energies.

TABLE VII Computational Performance for Two Methods

Method	Total restored load (kWh)
Method	Average value	Worst-case value
Proposed method	84417	77308
Deterministic optimization	83935	76184

E. Impact of Initial Capacity of Stocked Batteries

The initial capacity of stocked batteries in NBCSS is influenced by EV behavior. In this subsection, we delve into the impact of the initial capacity of stocked batteries on service restoration. According to the number of initial batteries with different SOCs in NBCSS, the initial capacity ratio can be easily calculated. Consequently, three different initial capacity ratios (0.3, 0.5, and 0.8) are designated for comparative analysis, representing low, medium, and high initial capacity ratios, respectively. The total restored load and generation cost under different ratios are listed in Table VIII. Notably, the total restored load remains largely consistent across the three ratios. However, there is a reduction in generation cost as the initial capacity ratio increases. This is mainly attributed to the GT in NBCSS, which can provide insufficient power, making the NBCSS more effective in supporting the service restoration of DS.

TABLE VIII Total Restored Load and Generation Cost Under Different Ratios

Initial capacity ratio	Total restored load (kWh)	Generation cost (¥)
0.3	85019	11353
0.5	85049	9027
0.8	85065	8065

F. Sensitivity Analysis on Number of Stochastic Scenarios

In this subsection, the sensitivity analysis is carried out to evaluate the impact of the number of stochastic scenarios on the optimization results. In addition to the 10 typical scenarios reduced from 1000 scenarios, 5, 15, and 20 typical scenarios are utilized for comparison. The comparison results of different numbers of stochastic scenarios are listed in Table IX. As the number of stochastic scenarios increases, the objective function value converges to ¥77671, accompanied by a gradual rise in computation time. Particularly, the objective function values vary much between stochastic scenario 5 and stochastic scenario 10. By comparison, the objective function values vary little between stochastic scenario 10 and stochastic scenario 20. These findings suggest that utilizing 10 typical scenarios strikes a balance between result accuracy and solution time.

TABLE IX Comparison Results of Different Numbers of Stochastic Scenarios

Number of stochastic scenarios	Objective function value (¥)	Solution time (s)
5	75275	512
10	77042	932
15	77393	1613
20	77671	4123

G. Computation Complexity Analysis on Number of Damaged Lines

The repair crew routing is a NP-hard problem, significantly amplifying the computational complexity of the proposed method. The more the number of damaged lines, the greater the computational complexity. Therefore, the solution time with different numbers of damaged lines is discussed, as shown in Table X. With the increase in the number of damaged lines, the solution time increases in a more than linear manner. Consequently, when dealing with a substantial number of damaged lines, certain measures such as pre-assignment of repair tasks [

16], must be taken to alleviate computation time.

TABLE X Solution Time with Different Number of Damaged Lines

Number of damaged lines	Solution time (s)
3	323
6	932
9	2362

H. Test on Modified IEEE 69-bus DS

As depicted in Fig. 12, the modified IEEE 69-bus DS [

29] is also simulated to further verify the proposed method. The NBCSS links to the DS by bus 8. One WP and two PVs are also placed in the DS. Besides, the DS possesses three remote-controlled switches. There are nine damaged lines in the DS. Three repair crews in two depot sites are responsible for repairing damaged lines. To accelerate the solution time, the repair crews 1 and 2 are arranged for damaged lines L1-L6, while the repair crew 3 is arranged for damaged lines L7-L9. The repair time of the nine damaged lines is listed in Table XI.

Fig. 12 Topology structure of modified IEEE 69-bus DS.

TABLE XI Repair Time of Damaged Lines for Three Crews

Crew	Repair time (hour)
Crew	L1	L2	L3	L4	L5	L6	L7	L8	L9
Crew 1	3	3	3	4	3	4
Crew 2	3	4	3	3	3	4
Crew 3							3	3	4

Under the given relevant parameters, the optimization problem is successfully solved. Table XII presents the routing of different repair crews, while Table XIII presents repair completion and available time of damaged lines. Besides, the restored load of DS at different time is also given in Fig. 13. As the damaged lines are sequentially repaired, the load of the DS gradually recovers. The results verify the effectiveness of the proposed method again.

TABLE XII Routing of Different Repair Crews

Crew	Sequence of repair
Crew 1	Depot site 1→L2→L1→L6→Depot site 1
Crew 2	Depot site 1→L4→L3→L5→Depot site 1
Crew 3	Depot site 2→L7→L8→L9→Depot site 2

TABLE XIII Repair Completion and Available Time of Nine Damaged Lines

Damaged line	Repair completion time (hour)	Available time (hour)
L1	6.7	8
L2	3.5	5
L3	6.9	8
L4	3.6	5
L5	10.2	12
L6	11.1	13
L7	3.5	5
L8	6.8	8
L9	11.1	13

Fig. 13 Restored load of DS at different time.

VI. Conclusion

This paper proposes service restoration considering NBCSS, repair crews, and network reconfigurations.

Furthermore, the stochastic optimization is employed to cope with the uncertainties of renewable energies, while CVaR measure is also considered for risk aversion. In the stochastic optimization model, copula theory is further adopted to capture the correlation between the output of adjacent renewable energies. The results of case studies indicate that the proposed method can effectively dispatch crews to repair damaged lines and NBCSS to support the DS. Besides, the coordination of the NBCSS, repair crews, and network reconfigurations can restore more load than no coordination at all. Moreover, the uncertainty risks of renewable energies are well addressed.

In this paper, the movement of repair crews is assumed to be within a normal transportation system. However, extreme events have a destructive impact on the transportation system. In future work, we will integrate the transportation system into the service restoration problem of DS.

Nomenclature

Symbol	——	Definition
A.	——	Indices and Sets
$Φ_{F}$	——	Set of lines travelled by crews
$Ω_{D}, Ω_{S}$	——	Sets of damaged lines and switchable lines
hs	——	Indices of $W_{i j}^{U} a n d W_{i j}^{D}$
$O_{l}$	——	Set of potential loops
s	——	Index of uncertain scenarios
$S_{j}^{B, d}$	——	Set of downstream buses connecting to bus j
t, τ	——	Indices of dispatching period T
$W_{i j}^{U}, W_{i j}^{D}$	——	Sets of upstream and downstream lines of branch ij
B.	——	Variables
$Φ_{ρ}$	——	Standard multivariate normal distribution
$Φ^{- 1}$	——	Inverse function of standard normal distribution
$ε_{i j, t}$	——	Binary variable indicating whether branch ij is closed at time t
$β_{i, c, t}, α_{i, c, t}$	——	Continuous variables about charging and discharging priorities of batteries
$δ_{i, c, t}, θ_{i, c, t}$	——	Binary variables about charging and discharging priorities of batteries
$σ_{c, t}, σ_{d, t}$	——	Binary variables about flag of charging and discharging of charging and discharging equipment (CDE)
$σ_{c h, t}, σ_{d i s, t}$	——	Binary variables about flag of charging and discharging of energy storage (ES)
$E_{e, t}$	——	Energy of ES
$f_{i, t}$	——	Binary variable indicating whether line i is repaired at time t
$N_{i, s, t}$	——	Quantity of batteries in the i^th state of charge (SOC) interval
$N_{i, c, t}, N_{i, d, t}$	——	Quantities of charging and discharging batteries in the i^th SOC interval
$N_{a l l, c, t}, N_{a l l, d, t}$	——	Total quantities of charging and discharging batteries
$P_{a l l, c, t}, P_{a l l, d, t}$	——	Total charging and discharging power of CDE in novel battery charging and swapping station (NBCSS)
$P_{c h, t}, P_{d i s, t}$	——	Charging and discharging power of ES
$P_{g, t}, Q_{g, t}$	——	Active and reactive power of gas turbine (GT)
$P_{t}^{N, L}, Q_{t}^{N, L}$	——	Restored active and reactive power of load in NBCSS
$P_{t}^{N, W P}, Q_{t}^{N, W P}$	——	Active and reactive power of wind power (WP) in NBCSS
$P_{t}^{N, P V}, Q_{t}^{N, P V}$	——	Active and reactive power of photovoltaic (PV) in NBCSS
$P_{j, t}^{N, e x}, Q_{j, t}^{N, e x}$	——	Exchanged active and reactive power moves from line j to line i
$P_{i j, t}^{D}, Q_{i j, t}^{D}$	——	Active and reactive power flow through branch ij
$P_{j, t}^{D, W P}, Q_{j, t}^{D, W P}$	——	Active and reactive power of WP at bus j
$P_{j, t}^{D, P V}, Q_{j, t}^{D, P V}$	——	Active and reactive power of PV at bus j
$R_{i, c}$	——	Binary variable indicating whether line i is repaired at time t
$T_{i, c}$	——	Time when crew c arrives at line i
$V_{i, t}^{}$	——	Voltage of bus i
$x_{j, i, c}$	——	Binary variable indicating whether crew c moves from line j to line i
$x_{g, t}, u_{g, t}, v_{g, t}$	——	Binary variables indicating on/off state of GT
$y_{i, t}$ , $y_{k (i j), t}$	——	Binary variables indicating whether line i and branch ij are available at time t
C.	——	Parameters and Constants
$λ_{e, m i n}, λ_{e, m a x}$	——	The minimum and maximum energy statuses
$η_{c h, e}, η_{d i s, e}$	——	Charging and discharging efficiencies of ES
$ε$	——	Error factor
$Δ t$	——	Time interval
$Β_{l}$	——	Number of lines of potential loops
$C^{S U}, C^{S D}$	——	Start-up and shut-down costs of GT
$C_{g}$	——	Unit generation cost of GT
$E_{e, r a t e}$	——	Rated capacity of ES
K	——	Number of SOC intervals
M_big	——	Large number
$N_{m a x}$	——	The maximum quantity of chargers in CDE
$P_{g, m i n}, P_{g, m a x}$	——	The minimum and maximum power of GT
$P_{f i x, c}$	——	Charging power of CDE for a single battery
$P_{f i x, d}$	——	Discharging power of CDE for a single battery
$P_{c h, m a x}$ , $P_{d i s, m a x}$	——	The maximum charging and discharging of ES
$P_{t}^{N, L, m a x}$ , $Q_{t}^{N, L, m a x}$	——	The maximum active and reactive power of load in NBCSS
$P_{j, t}^{N, e x, m a x}$ , $Q_{j, t}^{N, e x, m a x}$	——	The maximum exchanged active and reactive power between DS and NBCSS
$P_{j, t}^{D, L, m a x}$ , $Q_{j, t}^{D, L, m a x}$	——	The maximum active and reactive power of load at bus j
$P_{t, R E}^{N, W P}, P_{t, R E}^{N, P V}$	——	The maximum active power of WP and PV in NBCSS
$P_{i, t, R E}^{D, W P}, P_{i, t, R E}^{D, P V}$	——	The maximum active power of WP and PV in DS
$r_{i j}^{D}, x_{i j}^{D}$	——	Resistance and reactance of branch ij
$R_{U, g}, R_{D, g}$	——	Ramp-up and ramp-down limits of GT
$S_{m a x}^{N, W P}, S_{m a x}^{N, P V}$	——	The maximum capacities of WP and PV in NBCSS
$S_{i, m a x}^{D, W P}, S_{i, m a x}^{D, P V}$	——	The maximum capacities of WP and PV in DS
$S_{g, m a x}$	——	The maximum capacity of GT
$S_{U, g}, S_{D, g}$	——	Start-up and shut-down limits of GT
T	——	Dispatching period
$t_{g}^{U}, t_{g}^{D}$	——	The minimum up-time and down-time of GT
$t_{0}$	——	Initial departure time of crews
$t_{j, i, c}^{t r i p}$	——	Travel time between line j and line i of crew c
$t_{j, c}^{r e p}$	——	Repair time of line j of crew c
$V_{i, m i n}^{}, V_{i, m a x}^{}$	——	The minimum and maximum voltage limits of bus i
$V_{0}^{}$	——	Rated voltage magnitude

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